why does log a + log b = log (ab)

Let log a be some number A and log b be some number B

now the natural log of something is the equivalent of saying a=e^A and b = e^B

So a*b = e^A * e^B which by rules of indices

 = e^(A+B)

Therefore log(ab) = log(e^(A+B))

= A + B = log a + log b